The three methods (contiguity edges, edges + corner, k-nn) may be using similar neighborhoods for analysis. Depending on the layout of the polygons contiguity edges and edges + corner can produce very similar neighborhoods if there are few polygons that are only connected by a corner. Using edges + corners can have a big impact if you have a regular spaced grid (fishnet) as you double the neighborhood from 4 edges to 8 with edges + corners.
In this example polygon data set the majority of polygons are connected by edges
You can try to assess the number of neighbors each polygon is getting by using the Polygon Neighbors Tool it will create a table with a row for each polygon and its neighboring polygons. Using the Summary Statistics tool with a case field (group field) by the src_OBJECTID and counting the nbr_OBJECTID will produce a table with the number of neighbours for each polygon. If you create a bar chart using the number of neighbors you can see the distribution of number of neighbors. If the average distribution of number of neighbors is similar to the number of neighbors used in the analysis, it could help explain the similar results.
Table from Polygon Neighbors Tool
Results from Summary Statistics Tool
Chart showing distribution of number of neighbors